We just want to find lower half-plane pole-skipping points which also locate at Matsubara frequencies \(\mathfrak {w}_{n}:=\frac{\omega _n}{2\pi T}=-in\)  in this background. So we only consider the linear perturbation of axion field \(\chi (r)=\beta x+\bar{\chi }(r)e^{-i\omega t+i k_x x}\)  in this section. We choose the parameter \(\beta =0.1\)  which is small enough in the following. We take the equation of motion of axion field as a probe. Then we write down the equation of motion for axions [1]}, [2]}, [3]}
\(\nabla _{\mu }\big (Y(\phi )\nabla ^{\mu }\chi \big )=0.\) 
